Communication Using Chaotic Waveforms

ABSTRACT

Example communication systems and methods are described. In one implementation, a method receives a first chaotic sequence of a first temporal length, and a second chaotic sequence of a second temporal length. The method also receives a data symbol for communication to a destination. Based on the data symbol, the second chaotic sequence is temporally shifted and combined with the first chaotic sequence to generate a composite chaotic sequence. The first chaotic sequence functions as a reference chaotic sequence while the second chaotic sequence functions as a data-carrying auxiliary chaotic sequence.

RELATED APPLICATIONS

This application is a continuation, and claims priority to U.S. patentapplication Ser. No. 15/686,783, entitled “Communication Systems andMethods,” filed on Aug. 25, 2017, the disclosure of which isincorporated by reference herein in its entirety. That applicationclaims the priority benefit to U.S. patent application Ser. No.15/079,913, entitled “Communication Systems and Methods,” filed on Mar.24, 2016, the disclosure of which is incorporated by reference herein inits entirety. That application claims the priority benefit of U.S.patent application Ser. No. 13/708,822, entitled “Chaotic communicationsystems and methods,” filed Dec. 7, 2012, the disclosure of which isincorporated by reference herein in its entirety. That applicationclaims the priority benefit of U.S. Provisional Application Ser. No.61/568,037, entitled “Chaotic communication systems and methods,” filedDec. 7, 2011, the disclosure of which is incorporated by referenceherein in its entirety.

STATEMENT REGARDING FEDERALLY-SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under Contract No.W15P7T-11-C-H203 awarded by the U.S. Army. The government has certainrights in the invention.

TECHNICAL FIELD

The present disclosure generally relates to communication systems and,in particular embodiments, to digital communication systems and methodsusing chaotic signals as modulating waveforms.

BACKGROUND

Various communication systems are available for communicating databetween two locations. For example, spread-spectrum broadband digitalcommunication systems have long offered several advantages overnarrowband communication systems, including better immunity to noise,improved multipath rejection and, in the case of signals hidden in thenoise floor, a low probability of intercept. Additionally, code divisionmultiple access (CDMA) techniques allow multiple users to be supportedon the same bandwidth. Typical applications of these digitalcommunication techniques include cellular phones and the globalpositioning systems. However, these existing systems are oftenvulnerable to spoofing, jamming, and various techniques that mayintercept the communicated signals.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example chaotic communication system in accordancewith an embodiment.

FIG. 2 is a flow diagram of a method, in accordance with an embodiment,for generating and transmitting a chaotic data signal.

FIG. 3 is a flow diagram of another method, in accordance with anembodiment, for generating and transmitting a chaotic data signal.

FIG. 4 is a block diagram of a chaotic signal transmitting system inaccordance with an embodiment.

FIG. 5 is a block diagram of a chaotic signal receiving system inaccordance with an embodiment.

FIG. 6 is a flow diagram of a method, in accordance with an embodiment,for generating a chaotic sequence in fixed-point arithmetic using aCORDIC block.

FIG. 7 is a block diagram of a ChCM (Chaotic Code Multiplexing) signalgenerator in accordance with an embodiment.

FIG. 8 illustrates an example of the code tracking portion of a ChCMreceiver in accordance with an embodiment.

FIG. 9 is a signal composition diagram for a modified ChCM system inaccordance with an embodiment.

FIG. 10 illustrates an example matrix showing ChCM intervals and partialsums in accordance with an embodiment.

FIG. 11 illustrates an example addition algorithm used in a modifiedChCM system in accordance with an embodiment.

FIG. 12 illustrates an example of combining two or more chaoticsequences in order to increase the spectral efficiency of a chaoticcommunication system in accordance with an embodiment.

FIG. 13 illustrates an example of how correlation peaks of two or morechaotic signals can be used to carry communication symbols in accordancewith an embodiment.

FIG. 14 illustrates a flow diagram that describes a method for combiningtwo chaotic waveforms at a transmitter associated with a chaoticcommunication system in accordance with an embodiment.

FIG. 15 illustrates a flow diagram that describes a method for receivingand demodulating a combination of two chaotic waveforms at a receiverassociated with a chaotic communication system in accordance with anembodiment.

FIG. 16 illustrates a graphical plot that illustrates two possibledegrees of freedom used to increase spectral efficiency in accordancewith an embodiment.

FIG. 17 is a graphical plot that illustrates how contemporary signalmodulation techniques can be used in conjunction with chaotic modulationtechniques to further increase spectral efficiency in accordance with anembodiment.

FIG. 18 illustrates a block diagram that depicts a system for generatinga discrete-time frequency-modulated chaotic sequence in accordance withan embodiment.

FIG. 19 illustrates a block diagram that depicts a system for generatinga discrete-time phase-modulated chaotic sequence in accordance with anembodiment.

FIG. 20 illustrates a block diagram that depicts a system for generatinga discrete-time frequency-modulated chaotic sequence in accordance withan embodiment.

FIG. 21 illustrates a block diagram that depicts a system for generatinga discrete-time phase-modulated chaotic sequence in accordance with anembodiment.

FIG. 22 illustrates a diagram illustrating different methods of maskinginformation-carrying chaotic waveforms to increase the security of thecommunication system.

DETAILED DESCRIPTION

The systems and methods described herein use chaotic signals asmodulating waveforms to communicate data to a destination. The chaoticsignals replace, for example, conventional binary spreading codes asused in current spread-spectrum techniques. These described systems andmethods use a slice of a chaotic waveform, sampling and quantizing it,and repeating it periodically. The chaotic sequence can be generatedeither a priori and stored in a file that can be read, or generated inreal-time with a chaotic signal generator that uses, for example, theCORDIC (COordinate Rotation DIgital Computer) block in a programmablelogic device, such as a FPGA (field programmable gate array).

Spread-spectrum broadband digital communication systems offer severaladvantages over narrowband communication systems, including betterimmunity to noise, improved multipath rejection and, in the case ofsignals hidden in the noise floor, a low probability of intercept.Further, code division multiple access (CDMA) techniques allow multipleusers to be supported on the same bandwidth. Typical examples of thesedigital communication techniques are cellular phones and the GlobalPositioning System. These systems use binary spreading waveforms, andwhile they offer the advantages discussed above over narrowbandcommunication systems, they still have their limitations—for example, aGPS (global positioning system) signal is vulnerable to spoofing.Further, carrier and code clock regeneration techniques take advantageof the binary modulation schemes and can be used to intercept thesesignals.

Example chaos-based digital communication schemes are described herein.Chaotic signals are deterministic, reproducible signals with theproperties of noise. Chaotic signals are generated by nonlinear systems,either in the analog domain or digitally. These systems are verysensitive to initial conditions, with different temporal waveforms beingproduced by the same system with only a slight change in initialconditions. The frequency domain representation of these signalsresembles noise spectra, and they display strong autocorrelationproperties, with weak cross-correlations. These properties of chaoticsignals make them very desirable for use in digital communicationsystems, since they offer several improvements over spread spectrumsystems using binary modulation, such as lower probably of interceptionand better security.

The systems and methods described herein include one or more of thefollowing features:

Uses a slice of a chaotic sequence that is sampled and quantized, andrepeated periodically.

CORDIC block implementation of chaotic generator allows for real-timegeneration of chaotic sequence.

Does not need resynchronization signals to be sent periodically in caseof loss of communication link.

Software-defined chaotic transmitter and receiver pair does not needspecialized processing hardware.

Can be implemented using conventional clock generators; does not requirea high-precision clock (e.g., GPS or atomic clocks) at either thetransmitter or the receiver.

Capable of operating below the noise floor.

Secure communication system—signal or message cannot be decoded byanyone who does not have knowledge of the chaotic modulating signal.

High Data Rates.

Does not require or use a chaotic signal generator that runsindefinitely since the chaotic sequence is periodic.

FIG. 1 illustrates an example chaotic communication system 100, inaccordance with an embodiment. As shown, a signal generator 102 receivesa data signal from a data source 104, and receives a chaotic waveformfrom a chaotic waveform generator 106. In alternate embodiments, thechaotic waveform generator 106 is replaced with any system or devicecapable of providing a chaotic waveform to the signal generator 102. Insome embodiments, the chaotic waveform generator 106 is a storage devicethat stores one or more chaotic waveforms (or portions of chaoticwaveforms). The signal generator 102 generates a chaotic data signal,which includes the data signal as modulated by at least a portion of achaotic waveform. In some embodiments, the chaotic data signal isgenerated by multiplying the data signal and the chaotic waveform. Thechaotic data signal is provided to a transmitter 108, which transmitsthe chaotic data signal to a destination (e.g., a receiver 110 or othersystem at the destination).

The receiver 110 at the destination receives the chaotic data signal andprovides the signal to a signal processor 112. The signal processor 112extracts the data signal from the chaotic data signal based on thechaotic waveform data 114. The chaotic waveform data 114 used by thesignal processor is the same as (or is a representation of) the chaoticwaveform used by the signal generator 102 that generated the chaoticdata signal (e.g., the chaotic waveform generated by chaotic waveformgenerator 106). The extracted data signal is then provided to anothersystem or component for processing.

FIG. 2 is a flow diagram of a method 200, in accordance with anembodiment, for generating and transmitting a chaotic data signal. Themethod 200 receives a portion of a chaotic waveform at 202 and samplesthe received chaotic waveform at 204. The sampled waveform is quantizedat 206 and stored at 208 for future access by a signal generator orother component/system. Quantizing the waveform refers to the process ofmapping a large set of input values to a smaller set, which is part ofthe waveform digitization process. This digitization process includessampling the waveform in time and quantizing the waveform to account forthe effects of the analog-to-digital converter, discussed herein. Themethod 200 then receives a data signal that is to be communicated to adestination (e.g., via a wireless communication link) at 210. Thereceived data signal is modulated using the quantized chaotic waveformat 212 and the modulated data signal is transmitted to the destinationat 214. Additional details regarding the generation and transmitting ofchaotic data signals are provided herein.

FIG. 3 is a flow diagram of another method 300, in accordance with anembodiment, for generating and transmitting a chaotic data signal. Themethod 300 receives a portion of a chaotic waveform at 302 and samplesthe chaotic waveform at 304. A portion of the sampled chaotic waveformis selected at 306 for applying to received data signals. The selectedportion of the sampled chaotic waveform has a temporal length (alsoreferred to as a “periodic interval”). The method 300 continues byquantizing the selected portion of the sampled chaotic waveform at 308.The method 300 continues as a data signal is received at 310. Thereceived data signal is to be communicated to a destination. Thereceived data signal is modulated at 312 using the quantized portion ofthe chaotic waveform and the modulated data signal is transmitted to thedestination at 314.

FIG. 4 is a block diagram of a chaotic signal transmitting system 400 inaccordance with an embodiment. In the embodiment of FIG. 4, a FPGA 402is coupled to a clock generator 404 that provides a clock signal to theFPGA 402. The FPGA 402 includes a digital multiplier 406 that receivessignals from a direct digital synthesizer 408 and a digital data source410. The FPGA 402 also includes a 4-bit chaotic signal source 412 thatprovides a chaotic signal to the direct digital synthesizer 408 toaccount for various modulation schemes, such as amplitude modulation,frequency modulation, and phase modulation. The output of digitalmultiplier 406 is a 10-bit digital chaotic signal including a code and acarrier. The 10-bit digital chaotic signal is provided to a DAC(digital-to-analog converter) 414. Additionally, the FPGA 402communicates the clock signal received from the clock generator 404 tothe DAC 414. The DAC 414 communicates the converted signal to an IF(intermediate frequency) amplifier 416, which communicates the amplifiedsignal to an upconverting mixer 418. The upconverting mixer 418 receivesa signal from an RF (radio frequency) generator 420. The output of themixer 418 is communicated to an image reject filter 422. The output ofimage reject filter 422 is communicated to an RF amplifier 424, whichamplifies the upconverted chaotic signal to provide the necessary signalpower prior to RF transmission to a destination device.

An example implementation of the chaotic signal transmitting system 400communicates wirelessly in the L-band (950-1950 MHz). In this exampleimplementation, the clock generator 404 generates a master clock signalfor the chaotic signal transmitting system 400. This master clock signalhas a frequency of 52.0833 MHz. The FPGA 402 is the source of thedigital chaotic signal. The 4-bit chaotic signal source 412 produces adigitized chaotic signal that clocks at one fourth the master clock rate(i.e., the clock signal generated by the clock generator 404). At a rateof four master clock samples per chaotic sequence sample, the resultantchaotic signal sequence rate is 13.028 MHz. Also, the chaotic sequencerepeats every 125 samples, and assuming that the digital data modulatesthe chaotic signal at the repeat rate, this gives a resultant data rateof 104.167 kbps.

In the example implementation of chaotic amplitude modulation, thedirect digital synthesizer generates an 8-bit wide digital carrier sinewave that multiplies the chaotic signal (chaotic amplitude modulation).The digital carrier sine wave has a frequency of 15 MHz. This signalupconverts the spectrum of the chaotic signal, and reduces the high-passeffects of the digital-to-analog converter on the chaotic signal, whichis sampled and held. If the chaotic signal is directly input to the DAC,then the lower frequencies get distorted since the output of the DAC istransformer-coupled. This results in a nonlinear transient response inthe flat portions of the chaotic signal. Upconverting the signal in thedigital domain itself minimizes the effect of the DAC on the lowfrequency components of the signal. The output composite code-carrierdigital data is 10-bit. However, the multiplication of the code andcarrier gives a 12-bit number. This 12-bit number has its lower 2 bitstruncated to give a 10-bit output. This procedure is performed toincrease the fidelity of the multiplication process.

The digital data from the data source 410 is a stream of ones andzeroes. If the value of the data bit is one, then the 10-bit digitalchaotic signal is not modified. However, if the value of the data bit iszero, then the 10-bit digital chaotic code-carrier signal is multipliedby −1.

In the example implementation, the DAC 414 is a 10-bit, 100 MSPS capabledevice. The DAC 414 receives as input the system clock of 52.0833 MHzand the 10-bit digital chaotic signal described above. The IF amplifier416 conditions the signal output by the DAC 414 to drive theupconverting mixer 418. The RF generator 420 outputs a frequency of1.9375 GHz to the upconverting mixer 418. The upconverting mixer 418receives the output analog chaotic signal from the DAC 414 which iscentered at a 15 MHz analog frequency and mixes it with the 1937.5 MHzsinusoidal signal from the RF generator 420 to produce two versions ofthe chaotic signal—the first is a high-side upconvert, giving an outputfrequency of 1922.5 MHz and the second is the low-side upconvertfrequency of 1952.5 GHz. Both versions of the chaotic signal areprovided to image reject filter 422. The image reject filter 422 is apassive filter that suppresses the higher image frequency and passes thedesired upconverted signal to an antenna, which transmits theupconverted chaotic signal in the L-band.

FIG. 5 is a block diagram of a chaotic signal receiving system 500 inaccordance with an embodiment. In the embodiment of FIG. 5, an RFamplifier 516 receives a chaotic signal from an antenna and amplifiesthe chaotic signal in the RF band prior to downconversion. Adownconverting mixer 502 receives the amplified signal from the RFamplifier and receives an RF signal from an RF generator 504. Thedownconverting mixer 502 is coupled to an IF amplifier 506, which iscoupled to an ADC (analog-to-digital converter) 508. The ADC 508receives a master clock signal from a clock generator 510. Two outputsignals are provided from the ADC 508 to a FPGA 512: the master clocksignal (generated by the clock generator 510) and a 10-bit digitalchaotic signal. The FPGA 512 includes one or more chaotic signalacquisition and tracking algorithms 514 that extract a digital datasignal from the 10-bit digital chaotic signal.

An example implementation of the chaotic signal receiving system 500communicates wirelessly in the L-band (950-1950 MHz). In this exampleimplementation, the RF generator 504 drives one of the two inputs of thedownconverting mixer 502, which is used to mix the input RF signal at1922.5 MHz with the locally generated 1731.25 MHz signal to get adownconverted baseband center frequency of 191.25 MHz. The IF amplifier506 provides the necessary amplification to the downconverted signal inorder to drive the ADC 508. The clock generator 510 generates the masterclock for the chaotic signal receiving system 500. In this particularexample, the master clock has a frequency of 52.0833 MHz, which is thesame as the chaotic signal transmitting system 400 discussed above.

The ADC 508 is used to digitize the chaotic signal at baseband fordigital processing by the FPGA 512. The ADC outputs the 52.0833 MHzclock along with the 10-bit digitized chaotic data to the FPGA 512. TheFPGA 512 executes the acquisition and tracking algorithms in real-time,and demodulates the chaotic signal to generate the digital data output.

The design of the chaotic digital communication system is based on achaotic spreading sequence that is generated by an a priori generateddata file that is programmed into the FPGA itself. The discrete-timechaotic function used in some embodiments is based on the cubic map,given by the discrete-time recursive equation:

x _(k+1)=4x _(k) ³−3x _(k),  (1)

where different initial conditions x₀ give different chaotic timeensembles, where

|x₀|<1. An alternative to the algebraic equation (1) is to representthis cubic map by a trigonometric equation, where, by substituting:

x _(n)=cos ϕ_(n),  (2)

and then applying the triple angle formula, the result is:

x _(n+1)=cos ϕ_(n+1)=4 cos³ϕ_(n)−3 cos ϕ_(n)=cos 3ϕ_(n).  (3)

Then, the equation for the cubic map can be expressed by the followingtrigonometric equation:

cos ϕ_(n+1)=cos 3ϕ_(n),  (4)

which can be written as a recursion based on an initial value of ϕ atn=0 as:

cos ϕ_(n)=cos 3^(n)ϕ₀.  (5)

Thus, the sequence {cos ϕ_(n)}={cos 3^(n) ϕ₀}, for some ϕ₀ forms thebasis for a discrete-time trigonometric chaotic sequence generator.Further, by changing the multiplier on the angle from 3 to other valuessuch as 5, 7 and so on gives additional functions (maps) that can beused to generate chaotic time sequences in discrete-time. Theseproperties of trigonometric chaotic generating functions are used as thebasis for the fixed-point chaotic sequence generating algorithmdiscussed below

Fixed-Point Discrete-Time Chaotic Sequence Generator Using the CORDICBlock

From a computational perspective, the equations discussed above that areused to generate the chaotic time series use floating-point arithmeticto perform the computations. One embodiment of the discrete-time chaoticgenerator using floating point processing is accomplished by using anembedded processor on an FPGA. In alternate embodiments, thetrigonometric chaotic generating function can be modified to work in afixed-point (FPGA) environment, using the CORDIC block.

In some embodiments, computation is performed in fixed point; the inputangle as well as the output cosine function are in FixN_Q format. Inorder to generate the chaotic sequence, there are two parameters chosen:an initial condition and a trigonometric multiplier (equal to 3 for thecubic map). Then, at each iteration, the initial condition isrecursively multiplied by the trigonometric multiplying factor which isthen input to the CORDIC block. The time series output of the CORDICblock is the discrete-time chaotic sequence. By clocking the CORDICblock at a fraction of the master system clock (e.g., at a quarterrate), the sample-and-hold algorithm can be implemented, yielding theCDSS-SH (Chaotic Direct-sequence Spread Spectrum-Sample and Hold)chaotic sequence, which takes advantage of the early-late-promptdiscriminator function in order to implement the chaotic code trackingloop. While implementing the chaotic sequence generator in fixed-point,the recursive multiplication by the trigonometric multiplier results inthe value of the input to the CORDIC block growing with time. This canbe a problem for both fixed and floating point implementations. In orderto work around this, the input angle is constrained to be in theinterval (−π, π), and if the angle increases or decreases to a valuethat lies outside this interval, the value of π in the appropriateFixN_Q format is subtracted or added from the angle (depending on thesign) until the angle lies within the required interval. FIG. 6 is aflow diagram of this example method for generating a chaotic sequence infixed-point arithmetic using a CORDIC block.

The systems and methods discussed herein generate discrete-time chaoticsignals using fixed-point representation in devices such as FPGAs. Theadvantage of generating chaotic sequences in this manner allows forgreater flexibility as compared to the generation of such sequencesbased on text files that are loaded into the FPGA at program time. Also,it is possible that the fixed-point arithmetic implementation of thesechaotic sequence generators will be faster and less complex than thecorresponding implementations in floating-point arithmetic. The dyadformed by the initial phase angle and the trigonometric multiplier asinitialization variables can also allow for the reconfiguration andreprogramming of such chaotic generators in the field, and also allowfor “rolling codes” that allow the chaotic sequence generator to bereinitialized with a new code each time the system is turned on. Thisallows for greater security and immunity from eavesdropping. Increasingthe bit resolution on both the input and the output of the chaoticgeneration block allows for greater cipher strength and reducedprobability of interception and detection, while also giving a largerensemble of signals that can be generated using this technique. Due tothe finite-precision offered by the fixed-point format, the number ofpoints that can be generated by such chaotic generators is ultimatelylimited, and the chaotic sequences can exhibit periodicities. The sameis true for sequences generated using floating point processing, onlythat the time period of the corresponding orbits are much longer. Forfinite-length chaotic slices used as spreading sequences this is not alimitation, as different uncorrelated slices from the same chaoticsequence can be used for different receivers. Similarly, differenttrigonometric configurations of the CORDIC block in the FPGA (i.e., sineand tangent function generators) can be used to generate discrete-timechaotic sequences.

Example Chaotic Digital High-Speed Data Communication System

Certain systems and methods described herein use a single chaoticspreading code sequence that is sampled and quantized. This slice ofchaotic code is periodically repeated on a continuous basis, and thedigital data to be transmitted modulates the sign of the chaoticspreading sequence. This system, originally configured to transmitdigital data at a rate of 125 kbps, is currently configured to transmitdata at a rate of 694 kbps. The technique used to increase the datarate, i.e. by reducing the number of samples per code chip along withdecreasing the length of the integrate-and-dump sequence (therebydecreasing the repetition period of the chaotic sequence slice), islimited in the fact that the spreading sequence length cannot bearbitrarily decreased to a small value without severely compromising thesecurity features of the system while also greatly increasing theprobability of data bit errors. Although a sample rate increase can beused to increase the data rate, there is also an upper limit to how muchthe sample rate can be increased while keeping practical considerationslike system size, power consumption and weight in mind.

In order to work around these shortcomings, the following technique ispresented that allows for an increase in the data rate while maintaininga reasonable limit on system resources. This technique involves the useof two coexistent chaotic code spreading sequences. The first is a pilotcode sequence, which is acquired and tracked in a conventional manner.The data modulation technique, i.e. BPSK, can be introduced on thispilot code sequence, which allows it to carry additional information.The second coexisting code sequence is one out of a bank of N distinctchaotic code sequences. In the simplest realization of the system, eachsequence out of this bank is mapped to a binary symbol (which translatesto a binary data word of appropriate length), and depending on the inputdata word, the appropriate code sequence (referred to herein as theauxiliary chaotic code sequence) is selected and transmitted along withthe pilot chaotic code sequence. At the receiver, a standard set of codeand carrier tracking loops decode and track the pilot code sequence. Atthe same time, a complete set of replica sequences that are the same asthe auxiliary code sequences are simultaneously integrated-and-dumpedwith the input signal, and the signal with the largestintegrate-and-dump value corresponds to the transmitted auxiliary codesequence. The successful detection of this auxiliary code sequenceallows the receiver to appropriately select the corresponding binarydata word for output. Additional variations in this technique furtherallow for the increase in the data rate, as discussed herein. This datacommunication technique is referred to herein as Chaotic CodeMultiplexing (ChCM).

The basic ChCM signal is constructed using the correlation properties ofchaotic signals. These signals have strong autocorrelation propertiesand weak cross-correlations. Thus, it is possible for multiple chaoticsignals to coexist on the same bandwidth with minimum interference witheach other. The ChCM signal is derived as a summation of two signals: apilot chaotic code signal and an auxiliary chaotic code signal selectedfrom a bank of N distinct candidate signals. Each signal can beindependently correlated with its replica at the receiver with minimalinterference. In some embodiments, both signals are slices of chaoticsignals of equal length and repetition period. The pilot chaotic code isthus a fixed signal that does not change, while the auxiliary chaoticcode is variable and depends on the input digital data word. FIG. 7illustrates an example of a block diagram of a ChCM signal generator.

The pilot chaotic code signal as well as the multiplexed auxiliarychaotic signals in the respective bank are shown in FIG. 7. In aparticular example, there are N auxiliary chaotic signals in the bank,where N, for convenience, is selected to be a power of 2, such as:

N=2^(M),  (6)

where M is an integer number. Now, from binary arithmetic, this bank ofN signals can uniquely support the entire number set represented by theM-bit wide binary word. In other words, each auxiliary chaotic sequenceis mapped to a M-bit binary symbol. For example, if N=32, then thecomplete number set corresponding to 5-bit binary words can becompletely represented by this set of N codes: signal 1 represents thebinary word 00000, signal 2 represents the binary word 00001, and so on,until the signal 32 represents the binary word 11111. If the basic datarate supported by the basic system that uses only the pilot chaotic codesequence is D, then the ChCM system allows for a net data rate of MD.For example, for a 125 kbps system, using a 32-signal bank increases thedata rate by a factor of 5, to 625 kbps. Using larger banks of distinctchaotic signals will allow further multiplicative increases in datarates.

FIG. 8 illustrates an example of the code tracking portion of a ChCMreceiver in accordance with an embodiment. The ChCM receiver of FIG. 8is derived from the basic CDSS (Chaotic Direct-sequence Spread Spectrum)receiver. The basic acquisition and code/carrier tracking loops functionin the same way for the pilot chaotic code sequence. Once acquisition iscomplete and the locally generated pilot code is synchronized with theincoming signal, a second set of N parallel integrate-and-dump loopssimultaneously correlate the input signal with each of the signals inthe auxiliary chaotic signal bank. The output of the Nintegrate-and-dump blocks are evaluated at the end of everyintegrate-and-dump interval. Due to the correlation properties ofchaotic signals and the fact that the transmitted chaotic signals arephase-aligned and have the same time-period, only the auxiliary chaoticsignal that is present in the received waveform will produce a largevalue at the output of the respective integrate-and-dump block, whileother blocks will show small values. Thus, a peak-seeking algorithm canbe used to determine which output is the maximum, and upon finding themaximum peak in the output sequence, the corresponding M-bit binary wordcan be mapped to the output.

Modified ChCM Communication System for Higher Data Rates

FIG. 9 is a signal composition diagram for a modified ChCM system inaccordance with an embodiment. The following embodiment allows forhigher data rates from the ChCM system without needing a large number ofauxiliary code sequences. This enhancement utilizes the temporalcharacteristics of chaotic signals to design a modifiedtransmitter/receiver structure as described below.

The modified ChCM signal structure is still based on the combination ofmultiplexed auxiliary code signals and a pilot chaotic code sequence.The pilot chaotic code sequence in this modified design is unchangedfrom the basic system—it is the multiplexing of the auxiliary chaoticsignals that is modified. In this embodiment, the time period of thepilot signal (which is also the time period for the integrate-and-dumproutine) is divided into L equal partitions. During the time intervalcorresponding to each partition, a slice of any one of the N auxiliarycode sequences is inserted into the output stream, and the combinationof slices inserted into the output stream depends on the input binaryword. Thus, in each of the L intervals, there are N possible code slicesthat can be inserted, giving a total of N^(L) combinations of outputauxiliary chaotic code sequences. As before, if N is a power of 2, thenfrom equation (6), the widest binary word that can be supported by thesystem has a width

W=log₂(N ^(L))=L log₂(2^(M))=LM.  (7)

Thus, the division of the interval into L intervals increases the datarate of the basic ChCM system described above by a factor of L.

In a particular example, let N=2 (i.e. two auxiliary chaotic codesequences) and let L=5. For the simplest ChCM system, two symbols can betransmitted using this example: either a 1 or a 0, for an equivalentword width of 1 bit (i.e., M=1). If L=5 intervals are introduced and ineach interval, then it is seen that in each interval, there are twopossibilities—either a slice of the auxiliary chaotic signal 1 isintroduced or a slice of the auxiliary chaotic signal 2 is introduced.For five intervals, the total number of combinations are 2*2*2*2*2=32possibilities, giving a word width of log₂2⁵=5 bits=LM. This auxiliarysignal that is composed of a combination of slices of independentauxiliary chaotic sequences is combined with the pilot signal andtransmitted.

The complexities introduced into the modified ChCM signal structure arealso reflected in the receiver. However, the resulting system is lesscomplex for a given bit rate than the simplest ChCM receiver design,which would require a larger number of auxiliary chaotic code sequences.The basic structure of the ChCM receiver remains the same, with the samebank of auxiliary chaotic code sequences. As before, the pilot chaoticcode sequence is independently tracked by a separate set of trackingloops, while the input chaotic signal is pointwise multiplied by each ofthe auxiliary chaotic signals from the signal bank. The main differencebetween the two systems is in the way the integrate-and-dump values areprocessed. After every time interval T/L, where T is the period of thepilot and the auxiliary code signals, the values of each of theintegrators (partial sum) is stored in a separate register, and theintegrators are reset. Although the integrators are reset, the chaoticcode generators are not reset, and are allowed to run until the end ofthe interval T. At the end of each interval T, there are LN registerscontaining partial correlations of the input signal with slices oflength T/L of all the auxiliary codes. By grouping the partial sums fromthe integrators from the same time interval into a separate bank, thereare L banks of partial sums as shown in a matrix form in FIG. 10. If thedyad ij denotes the cells in FIG. 10, then i corresponds to the partialsum corresponding to the i^(th) auxiliary chaotic code sequence, and jcorresponds to the j^(th) interval. Using different combinations ofthese partial sums provides different correlation values. Specifically,different combinations of L entries from this matrix are taken, one fromeach column, exhausting all possible combinations for all N auxiliarychaotic code sequences. Examples of different possible combinations fora case with four intervals and two auxiliary chaotic code sequences are:

-   -   (11, 12, 13, 14),    -   (11, 12, 13, 24),    -   (11, 12, 13, 34),    -   (21, 32, 43, 14),        and so on. Each combination of four values is added together to        give a correlation value. If the sequence of partial sums        matches the code sequence on the received auxiliary chaotic code        signal, then that correlation value will be the maximum value.        In this example, if the received auxiliary chaotic code signal        corresponds to the sequence (11, 12, 13, 14), then maximum        correlation occurs for the corresponding locally generated code,        and the described systems and methods detect the presence of        this sequence.

Although a peak-seeking algorithm will be able to determine the maximumpeak, the described systems and methods consider the fact that therewill also be smaller peaks corresponding to partial correlations. Thus,for the above received sequence, the partial sum sequence, such as (11,12, 13, 24), at the receiver will also produce a value that isapproximately 75% of the peak value. On the other hand, a sequence suchas (21, 22, 23, 34) at the receiver will produce a very small (i.e.,uncorrelated) value.

Modified ChCM Receiver Algorithm

As discussed above, at the end of each full integrate-and-dump routine,all different combinations of partial sums based on selections from eachcolumn are to be added. For N auxiliary chaotic code sequences and Lintervals, there are N possible selections from each column, which givesNL combinations of sums (of L values each) that are to be implemented.One technique is to “brute force” implement all such additions. Analternate technique, discussed below, reduces the number of additionsrequired by the algorithm. This alternate algorithm reduces the sums oflength L into smaller summations computed on-the-fly as the partial sumsfrom the integrate-and-dump routine are computed. For example, considera simple example with N=2 and L=3. To compute all possible relevantcombinations of partial sums without optimization, it would require 8addition combinations in total, each of 3 elements, giving a total of 16addition operations. To optimize the computation of the correlationpeak, it is noted that the total integration routine is composed ofthree smaller integrations. The first integration routine provides twopartial sums, referred to as S₁₁ and S₁₂. Similarly, the secondintegration routine provides two more partial sums, referred to as S₂₁and S₂₂. These sums are combined in the following manner to get thefollowing sums:

S ₁₁₁₁ =S ₁₁ +S ₂₁,

S ₁₁₂ =S ₁₁ +S ₂₂,

S ₂₁₁ =S ₂₁ +S ₁₂,

S ₂₁₂ =S ₂₁ +S ₂₂.

These four sums are further combined with the results from the thirdintegration routine, S₃₁ and S₃₂, in the following manner:

S ₁₁₁₁ =S ₁₁₁ +S ₃₁,

S ₁₁₁₂ =S ₁₁₁ +S ₃₂,

S ₁₁₂₁ =S ₁₁₂ +S ₃₁,

S ₁₁₂₂ =S ₁₁₂ +S ₃₂,

S ₂₁₁₁ =S ₂₁₁ +S ₃₁,

S ₂₁₁₂ =S ₂₁₁ +S ₃₂,

S ₂₂₁₁ =S ₂₁₂ +S ₃₁,

S ₂₂₁₂ =S ₂₁₂ +S ₃₂.

The eight signals computed above are then evaluated by the peak-seekingalgorithm, and the largest-valued signal is taken as the correspondingchaotic sequence in the received chaotic signal. If intermediate valuesare computed continuously, the total number of additions are reduced. Inthis case, only 12 addition operations are necessary, reduced from 16operations as discussed above. More complex signal structures (i.e.,larger values of N and L) will allow for a greater reduction in thenumber of adders required, thereby resulting in reduced systemcomplexity. Additionally, the breaking down of the single additionoperations in the non-optimized algorithm into several cascaded additionoperations does not introduce latency into the system. In the formercase, all additions are computed at the end of the mainintegrate-and-dump routine, while in the latter case, the additions arecomputed continuously, culminating at the end of the main integrationroutine. Thus, both processes end at the same time, with the formerprocess being a one-step process at the end of the main integrationroutine and the latter process being a continuous flow through the mainintegration routine. FIG. 11 illustrates the above-described exampleaddition algorithm used in a modified ChCM system. The example of FIG.11 shows a total of 12 addition operations.

Additional Enhancements to the ChCM Algorithm

In some embodiments, additional enhancements are made to the ChCMalgorithm that help increase the data rate while introducing differentlevels of complexity into the system. A few example enhancements arediscussed below.

BPSK Enhancement

In some chaotic digital communication systems, a BPSK (binaryphase-shift keying) data stream is used to modulate the sign of thechaotic signal. At the receiver, the polarity of the correlated symbolgives the value of the data bit (1 or 0). Similarly, in the modifiedChCM algorithm, BPSK modulation can be added on both the pilot chaoticcode signal as well as the auxiliary chaotic code signals. The decodingof the data bit on the pilot chaotic code is done using a standardmethod for chaotic demodulation. In the case of the auxiliary chaoticsignal, the partial sums are computed and recombined as before. The onlyadditional steps required before invoking the peak seeking routineare: 1) determine the sign of the correlated symbol (positive ornegative), and 2) compute the absolute value of all the correlationswhose values are negative. Then, when the sequence with the largestcorrelation is computed based on the absolute (positive) values of allthe sequences, the sign of that sequence computed in step (1) adds anadditional bit to the width of the binary output word. Thus, introducingBPSK modulation to the pilot and auxiliary chaotic codes allows for anaddition of a total of two more bits to the binary word width, with aminimal increase in system complexity.

Generalized Code Slice Insertion Modification

In the enhanced ChCM system discussed herein (e.g., FIG. 9), the sliceof code inserted into a particular time interval was selected from aparticular bank of signals. Thus, as shown FIG. 9, the chaotic slice forthe i^(th) interval is selected only from the i^(th) column in theauxiliary signal bank. This algorithm can be generalized further byallowing the insertion of a chaotic slice from any column into anyinterval. This allows increasing the number of code sequencecombinations from N^(L) to (NL)^(L). Once again, if N=2^(M), then thewidth of the binary word is log₂((2^(M)L)^(L))=L log₂L+ML. Forconvenience, set L=2^(K), so that the expression for the binary wordwidth is equal to L(K+M), where L is left in its original form as amultiplication factor. In this way, the bit rate can be increasedfurther, with the tradeoff being an increase in system complexity. Thetransmitter will have to have a modified architecture to insert thechaotic sequences from different intervals into a current interval,while the receiver will have to run more parallel integrate-and-dumproutines to account for the selection of additional sequences present ineach interval.

Using Multiple Auxiliary Chaotic Code Sequences

Another way to increase the data rate of the ChCM system is to use morethan one auxiliary chaotic code sequence simultaneously along with thepilot chaotic code sequence. In other words, the pilot chaotic codesequence is transmitted along with two or more auxiliary chaotic codesequences. Then, if there are N auxiliary chaotic code sequences in thebank, out of which n auxiliary chaotic code sequences are transmitted ateach time, then even without the multiple interval partitioning, thisgives C^(N) _(n) combinations of auxiliary chaotic code sequences, where

C ^(N) _(n) =N!/(N−n)!n!

For example, if there are four auxiliary chaotic code sequences (N=4),and two sequences are transmitted at one time, this gives 12 possiblecombinations of binary words. At the receiver, the two maximum signalpeaks are selected, which is mapped to the appropriate binary word. Byincreasing the number of auxiliary chaotic code sequences that aresimultaneously transmitted, the effective binary word width can beincreased. There is, however, an upper limit to how many chaotic codesequences can be simultaneously accommodated in a given bandwidth,since, after a certain point, the chaotic code sequences startinterfering with one another if the number of coexisting code sequencesis high.

The techniques of multiple intervals discussed above can also beamalgamated with this technique of multiple chaotic code sequences toachieve further increases in the effective bit width. For example,consider the simplest case of L intervals where each interval is filledby n slices of chaotic signals from the same column in FIG. 9 (n<N). Inthis example, each interval can be filled in C^(N) _(n) ways, and for Lintervals, producing (C^(N) _(n))^(L) possible combinations, with Llog₂(C^(N) _(n)) as the effective binary word width. The receiverarchitecture in this example is a modification of the architecturedescribed previously, tailored to detect multiple correlation peaks.

Spread spectrum signals have the property that they can be used for thedesign of digital communication systems even when the spread spectrumsignal is completely overwhelmed by noise. By carefully designing thespread spectrum signal integration interval, it is possible to recover aspread spectrum signal in extremely low signal-to-noise ratioenvironments. These systems have their limitations, however. Forexample, it is difficult to integrate beyond a certain time limit, dueto the possibility of a digital data bit transition. This limitation isnot seen with chaotic signals, as discussed below.

By way of example, consider a spread spectrum communication system thatuses a binary spreading sequence that takes values of ±1. Assume that inthe digital domain, there are N samples per integration period. Further,assume that the digital data is BPSK modulated, then it is well-knownthat such a system cannot integrate beyond N samples. If thesignal-to-noise ratio degrades sufficiently, then integration up to Nsamples might not be sufficient to extract the spread-spectrum signalfrom the noise floor, which results in a loss of the communication link.

Chaotic communication systems circumvent this problem by being able totrack in extremely low signal-to-noise environments by arbitrarilyincreasing the integration time. While the BPSK digital data is lost inthis process, ChCM techniques can be used to encode digital data intothe communication link. For example, consider a chaotic signal at theoutput of the ADC at a chaotic signal receiver:

s _(c) _(k) =d _(i) _(N) c _(k) sin(ωk)

where d_(i) _(N) =±1 is the binary digital data that is held constantduring each integration interval of length N, ω is the digitized carrierangular frequency, c_(k) is the chaotic spreading sequence, and k is thediscrete-time index. (Note that if binary spreading sequences were usedinstead of chaotic spreading sequences, then the system would usec_(k)=±1.) For the chaotic communication signal described above, supposethe signal-to-noise ratio drops drastically, to the point where a singleintegrate-and-dump period, N, is insufficient to acquire and track.Under normal circumstances, the system would break down, rendering thecommunication link fractured. Further, the integration time cantypically not be increased beyond N in the presence of the binary databit for this class of systems (including binary spread-spectrumsystems). In other words, integrating beyond a single bit interval isnot an option for contemporary spread-spectrum systems, since thecorrelation peak will not be consistent due to the potential change inthe polarity of the data bit d_(i) _(N) =±1.

As a salient feature of chaotic and non-binary communication systems,consider an approach to square the signal s_(c) _(k) as follows:

s _(c) _(k) ² =c _(k) ² sin²(ωk)

where the system considers the fact that d_(i) _(N) ²=1. Now, the systemcan correlate this signal with a second sequence derived from theoriginal chaotic sequence defined by:

σ_(k)

c _(k) ²

and thus:

$s_{c_{k}}^{2} = {\frac{\sigma_{k}}{2}( {1 - {\cos ( {2\omega \; k} )}} )}$

Now that the binary data bit has been stripped off, the signal can befiltered to remove the low frequency chaotic component, and the systemcan attempt to acquire and track the composite chaotic signal:

$\rho_{k} = {\frac{\sigma_{k}}{2}{\cos ( {2\omega \; k} )}}$

with arbitrarily long correlation lengths, using the newly definedsequence

$\{ \frac{\sigma_{k}}{2} \}$

at twice the original carrier frequency. Thus, if the signal-to-noiseratio drops (for example, if a broadband jammer is active), then insteadof abandoning the communication link, the integration length can beincreased to get a stronger correlation peak by switching to the newlydefined signal ρ_(k) as defined above by (possibly digitally) squaringthe received downconverted signal, and tracking can be continued, albeitat a much lower data rate. This is a “fail-safe” mode, where the systemis still robust to noise and jamming. Data can still be sent andreceived at this lower rate, and the communication link is still active.For example, ChCM can now be used in this situation to send data acrossthe link via a nonstandard protocol. When the signal-to-noise ratioimproves, the system can go back to its regular mode of operation. Inthe fail-safe mode, squaring the input signal also squares the noisesignal as well, and even longer integration lengths might be required tomaintain correlations, but the preservation of the communication link isstill possible. This represents a significant advantage of chaotic andnon-binary spreading sequences over existing BPSK spreadingsequences—the sequence ρ_(k) cannot be constructed if binary spreadingsequences are used, since the squaring of the signal strips off thebinary spreading sequence.

These described systems and methods, coupled with the ChCM technique,can be used to transmit data even in severely degraded signal-to-noiseenvironments. Alternatively, a system can be designed from the ground-upto take advantage of this property, operating below the noise floor in acovert manner as a secure, undetectable communication channel.

Particular examples discussed herein are related to chaotic waveforms.However, the systems and methods described herein are applicable toother types of spread spectrum waveforms, such as binary spread spectrumwaveforms. Additionally, the systems and methods discussed above arerelated to chaotic amplitude modulation. In other embodiments, thesesystems and methods are applicable to chaotic frequency modulation andchaotic phase modulation, as discussed below.

In the example of chaotic phase modulation, the phase of the chaoticsignal is modulated, as given by the following equation:

C _(PM) _(k)

sin(ωk+c _(k)+ϕ)

The phase-modulated signal (C_(PM)) typically possesses better spectralcharacteristics as compared to an amplitude modulated signal. Thereceiver design for the phase-modulated signal uses a spread-spectrumreceiver design methodology. In some embodiments, at the expense of aslightly poorer tracking signal-to-noise ratio, for the same chaoticwaveform, the corresponding amplitude-modulated chaotic receiver candemodulate and decode the phase-modulated signal.

In other embodiments, a frequency-modulated chaotic signal is generatedby directly modulating the frequency of a sinusoidal carrier. Themodulation equation for the frequency-modulated chaoticfrequency-hopping spread-spectrum (CFSS) signal is given by thefollowing equation:

C _(FM) _(k)

sin[(ω+c _(k))k+ϕ]

In this embodiment, the receiver design for the frequency-modulatedsignal uses a spread-spectrum receiver design methodology to demodulateand decode the chaotic frequency-hopping spread-spectrum signal.

FIG. 12 illustrates an example 1200 of combining two or more chaoticsequences in order to increase the spectral efficiency of a chaoticcommunication system in accordance with an embodiment. Chaotic sequencestypically are used to implement spread-spectrum communication systems.The general class of spread-spectrum communication systems, includingspread-spectrum communication systems that use binary spreadingsequences, has a disadvantage of poor spectral efficiency. In otherwords, the bandwidth required to transmit a spread-spectrum signal istypically much larger than the actual data rate supported by thespread-spectrum communication system. Spread-spectrum chaoticcommunication systems as implemented by the current state-of-the-arttypically suffer from poor spectral efficiency as well. Note that in thefollowing discussion, the terms “chaotic sequence,” “discrete-timechaotic sequence,” “chaotic waveform” or “discrete-time chaoticwaveform” may be used interchangeably.

The example 1200 illustrated in FIG. 12 is an example of a method thatuses multiple chaotic sequences to increase the spectral efficiency ofthe chaotic communication system, i.e. increase the data rate withouthaving to increase the sample rate or bandwidth of the system. TheChaotic Code Multiplexing scheme described herein is an example of amethod of increasing the spectral efficiency of a chaotic communicationsystem by transmitting different combinations of multiple chaoticsequences simultaneously. In some embodiments, a first chaotic sequencec1 1202 of a specific temporal length is combined with a second chaoticsequence c2 1204 of the same temporal length to generate a compositechaotic sequence cT 1206 that has the same temporal length as the firstchaotic sequence c1 1202. Composite chaotic sequence cT 1206 isconstructed with the initial, or starting, points of the first chaoticsequence c1 1202 and the second chaotic sequence c2 1204 are aligned. Insome embodiments, composite chaotic sequence cT 1206 is constructed bythe addition of the first chaotic sequence c1 1202 and the secondchaotic sequence c2 1204. In this embodiment, if this composite chaoticsequence cT 1206 is transmitted to a receiver and is demodulated by thereceiver by using, for example, identical replica sequences of the firstchaotic sequence c1 1202 and the second chaotic sequence c2 1204 byseparately using matched-filter correlators, the correlation peaksassociated with the correlation of the first chaotic sequence c1 1202and the second chaotic sequence c2 1204 will be aligned, being locatedat the initial points of the respective correlation functions. In otherwords, the difference between the starting point of the correlation peakassociated with the first chaotic sequence c1 1202 and the correlationpeak associated with the second chaotic sequence c2 1204 is 0. Thisalignment of the correlation peaks could correspond, for example, to adata symbol of “0” that is meant to be communicated to the receiver bythe transmitter.

Additional data symbols can be encoded into the composite chaoticsequence cT 1206 by introducing temporal phase shifts into the secondchaotic sequence c2 1204. In some embodiments, to introduce additionaldata symbols into the system, the starting point of the second chaoticsequence c2 1204 can be shifted by D samples in the time-domain, withthe residual samples of the second chaotic sequence c2 1204 beingwrapped around the starting point of the second chaotic sequence c2 1204to give a shifted second chaotic sequence c2 1210. Note that thetemporal length of the shifted second chaotic sequence c2 1210 is thesame as the temporal length of the second chaotic sequence c2 1204which, in turn, has the same temporal length of the first chaoticsequence c1 1202. Shifted chaotic sequence c2 1210 is combined with afirst chaotic sequence c1 1208 (which is the same as the first chaoticsequence 1202), to give a composite chaotic sequence cT 1212 that is thesame temporal length as the first chaotic sequence c1 1208. Now, if thecomposite chaotic sequence cT 1212 is transmitted to a receiver and isdemodulated by the receiver by using, for example, identical replicasequences of the first chaotic sequence c1 1208 and the second chaoticsequence c2 1204 (not the shifted second chaotic sequence c2 1210) byseparately using matched-filter correlators, the correlation peaksassociated with the correlation function associated with the firstchaotic sequence c1 1208 and the correlation function associated withthe shifted second chaotic sequence c2 1210 will not be aligned.Instead, there will be a difference of D samples between the startingpoint of the correlation peak associated with the first chaotic sequencec1 1208 and the correlation peak associated with the second chaoticsequence c2 1204. This alignment difference between the correlationpeaks could correspond, for example, to a specific data symbol that ismeant to be communicated to the receiver by the transmitter.

In a particular example, the first chaotic sequence c1 1202 has atemporal length of 1024 points. If the second chaotic sequence c2 1204is combined with the first chaotic sequence to form composite chaoticsequence cT 1206 then, as discussed above, the correlation peaksassociated with the first chaotic sequence c1 1202 and the secondchaotic sequence c2 1204 are aligned at the receiver with zero shiftbetween the correlation peaks, and this can correspond to a data symbolof “0.” If the second chaotic sequence c2 1204 is temporallyphase-shifted by one sample to get the shifted second chaotic sequencec2 1210 (D=1). Based on the above discussion, there will be a 1-sampledifference between the correlation peaks associated with the correlationfunction of the first chaotic sequence c1 1208 and the correlationfunction of the shifted second chaotic sequence c2 1210 at the receiver,and this can correspond, for example, to a data symbol of “1.” We canextend the above argument to a generalized temporal phase-shift of D=nsamples on the shifted second chaotic sequence 1210, where a shift ofD=n samples corresponds to a data symbol “n.” Since the temporal lengthof the first chaotic sequence c1 1202 and the temporal length of thesecond chaotic sequence c2 1204 for this example is 1024 points, it isstraightforward to see that D can go up to a maximum number of 1023, fora total of 1024 possible temporal phase-shifts (starting from a temporalphase-shift of 0). Thus, the data symbols associated with these 1024possible temporal phase-shifts can range from 0 to 1023, correspondingto a 10-bit wide binary data word. Therefore, the communication schemecan now receive 10-bit wide binary data words at the transmitter andtransmit these 10-bit wide binary data words to a receiver via temporalphase-shift coding via shifted second chaotic sequence c2 1210. In orderto see how spectral efficiency has increased, suppose the base data rateassociated with the first chaotic sequence temporal length is R. Usingthe temporal phase shifting method, the method has been able to increasethe data rate to 10R, without requiring any increase in bandwidth orsample rate. For example, if the sample rate associated with the firstchaotic sequence c1 is 1.024 MHz. For a 1024-point chaotic sequencelength, there is a base data rate of 1 kbps. Using a second chaoticsequence such as the shifted second chaotic sequence c2 1210 andemploying the temporal phase-shifting technique described above, thedata rate can be increased to 10 kbps.

In the above scheme, the first chaotic sequence c1 1202 (or 1208) servesas a reference chaotic waveform, also known as a pilot chaotic waveform,a reference chaotic sequence or a pilot chaotic sequence, that providesa reference correlation peak. In some embodiments, the reference chaoticsequence may be any one or a combination of an amplitude-modulatedchaotic waveform, a frequency-modulated chaotic waveform or aphase-modulated chaotic waveform as discussed above. The shifted secondchaotic sequence c2 1210 serves as an auxiliary chaotic waveform, ordata-carrying chaotic waveform, also known as an auxiliary chaoticsequence or a data-carrying chaotic sequence, where a shift in thestarting point of the shifted second chaotic sequence c2 1210 is used asa measure of the encoded data symbol. In some embodiments, thedata-carrying chaotic waveform may be any one or a combination of anamplitude-modulated chaotic waveform, a frequency-modulated chaoticwaveform or a phase-modulated chaotic waveform as discussed above. Fromthe above discussion, it is understood that if there are N allowedtemporal phase shifts in the shifted second chaotic sequence c2 1210,there is an increase in the data rate corresponding to log₂N over thebase data rate.

Another advantage of using chaotic waveforms for implementing the abovephase-shifting scheme is waveform diversity. In other words, a largeensemble of chaotic waveforms can be generated, and this property can beused to further improve the spectral efficiency of the system. Otherspread-spectrum sequences such as binary spreading sequences do not havethis kind of waveform diversity. In other words, a far greater number ofchaotic sequences of a given length can be generated as compared totheir binary counterparts. Another advantage that chaotic sequences haveover binary spreading sequences is that chaotic sequences can begenerated to be of any arbitrary length. Thus, while the above examplepresented a chaotic sequence length of 1024 points, this length is notrestrictive. The chaotic sequence length could be 1000 points, or 2000points without adversely affecting their quasi-orthogonal correlationproperties. Binary spreading sequences, on the other hand, do not havethis flexibility since they have to be of lengths that are functions ofpowers of 2. The ability to generate chaotic sequences of any arbitrarylength allows greater flexibility in the design of robust, spectrallyefficient communication systems by offering a larger number of degreesof freedom to the system designer.

Returning to the above example, select the shifted second chaoticsequence c2 1210 as a single waveform from a larger ensemble of, say 16chaotic waveforms, then for a total number of 1024 allowable shifts(including a zero phase shift), the net increase in data rate is:

log₂(1024×16)=14 times the base data rate.

In general, if there are N allowable phase shifts (including a zerophase shift) with an ensemble of M chaotic waveforms, of which onechaotic waveform is selected at a time for the shifted second chaoticsequence c2 1210, then the net increase in data rate is:

log₂(M×N)=log₂ M+log₂ N.

By using large ensembles of chaotic waveforms, these systems can greatlyimprove the spectral efficiency of this class of systems. Using adiverse set of chaotic waveforms in a manner described above also helpswith system security. By mapping a large ensemble of data symbols to alarge ensemble of chaotic sequences increases the diversity of thedata-mapping codebook and the corresponding encoding scheme used totransmit data. This makes it that much more difficult for an attacker orinterceptor to decipher the information being carried by a similarchaotic communications channel.

While the example 1200 illustrated in FIG. 12 shows two chaoticsequences—a first chaotic sequence c1 1202 and a second chaotic sequencec2 1204 being combined, it is possible to combine multiple sets ofshifted chaotic sequences. For example, we can have additional shiftedchaotic sequence c3, shifted chaotic sequence c4 and so on, included inthe composite chaotic sequence cT 1212. In general, for a singlereference chaotic sequence such as the first chaotic sequence c1 1202,if there are P data-carrying chaotic sequences transmittedsimultaneously with each data-carrying chaotic sequence beingindependently phase-shifted through N possible points, then it isstraightforward to show that net increase in the data rate over the basedata rate is given by

P log₂ N.

For example, if the system transmits 3 data-carrying chaotic sequencesin addition with the reference chaotic sequence, then for a total of1024 allowable temporal phase-shifts, the net increase in data rate is3×10=30 times the base data rate. Thus, the base data rate of 1 kbps asdescribed above can be increased to 30 kbps using 3 data-carryingchaotic sequences. Furthermore, if the P simultaneously-transmitteddata-carrying chaotic sequences are transmitted from an ensemble of Mchaotic sequences (M>P), then the net increase in data rate is given by:

log₂(N ^(P) C ^(M) _(P))=P log₂ N+log₂ C ^(M) _(P),

where C^(M) _(P) is the combinatorial operator, defined as

C ^(M) _(P) =M!/P!(M−P)!.

For example, if 3 data-carrying chaotic sequences are transmitted alongwith a reference chaotic sequence with 1024 phase shifts allowed and thedata-carrying chaotic sequences are selected from an ensemble of 16chaotic sequences, then the increase in data rate can be shown to be39.129 times the base data rate. Using the techniques outlined above,the spectral efficiency of chaotic communication systems can beimproved.

FIG. 13 illustrates an example 1300 of how correlation peaks of two ormore chaotic signals can be used to carry communication symbols inaccordance with an embodiment. In some embodiments, a reference chaoticsignal is transmitted along with a data-carrying chaotic signal asdiscussed above. At the receiver, a matched filter or similarcorrelation algorithm may be used to determine the starting point of thereference chaotic signal. An example correlation plot of the referencechaotic signal is represented by the plot 1302. An example correlationplot of the data-carrying chaotic signal is represented by the plot1304. The temporal phase shift between the correlation peak associatedwith the reference chaotic signal and the correlation peak associatedwith the data-carrying chaotic signal is shown by the plot 1302 and theplot 1304.

In this example, the temporal shift between the correlation peakassociated with the reference chaotic signal and the correlation peakassociated with the data-carrying chaotic signal is 1500 points, withthe correlation length of both the reference chaotic signal and thedata-carrying chaotic signal each being 2500 points. For the sake ofsimplicity, let us assume that out of the 2500 points associated withthe correlation length, only the first 2048 temporal phase shifts(including the zero phase shift) are used to encode data. Thiscorresponds to a log 2 2048=11-bit word width, where the binaryrepresentation corresponding to the zero temporal phase shift is00000000000, and the binary representation corresponding to a temporalphase shift of 2047 points is 11111111111. Using this encoding scheme,the temporal phase shift of 1500 points that is the difference betweenthe two individual correlation peaks 1302 and 1304 corresponds to a datasymbol represented as 10111011100 in a binary format. In this way, thetemporal phase shift between the reference chaotic signal and thedata-carrying chaotic signal is mapped to a data symbol belonging to ana priori determined alphabet or codebook. The receiver, having an apriori knowledge of this data symbol mapping, can then demodulate thereceived chaotic signal combination, determine the phase shift betweenthe reference chaotic signal and the data-carrying chaotic signal, anddecode the corresponding binary data symbol.

FIG. 14 illustrates a flow diagram that describes a method 1400 forcombining two chaotic waveforms at a transmitter associated with achaotic communication system in accordance with an embodiment. At 1402,the method generates a reference chaotic waveform of a specific temporallength. Example temporal lengths include 1024 point lengths or 2048point lengths as discussed in the examples above. Shorter temporallengths such as 125 points can also be used, as can longer temporallengths such as 20000 points. The temporal length is typically a designparameter that may be selected as a part of the system design process,and may be a function of parameters that include but are not limited torequired data rates, signal-to-noise ratio, bit error rates and so on.The advantage of using chaotic waveforms for the design ofspread-spectrum systems is that there is a great degree of flexibilityin choosing an appropriate waveform length to suit a particularapplication, since a chaotic sequence of any practically arbitrarylength can be generated without any constraints on the temporal lengthof the sequence.

Note that the term “reference chaotic waveform” is used synonymouslywith the term “reference chaotic sequence” described earlier. In someembodiments, the reference chaotic waveform may be one or anycombination of an amplitude-modulated chaotic waveform, afrequency-modulated chaotic waveform or a phase-modulated chaoticwaveform. Next, at 1404, the method generates an auxiliary chaoticwaveform of temporal length equal to that of the reference chaoticwaveform. Note that the term “auxiliary chaotic waveform” is usedsynonymously with the term “data-carrying chaotic sequence” describedearlier. In some embodiments, the auxiliary chaotic waveform may be oneor any combination of an amplitude-modulated chaotic waveform, afrequency-modulated chaotic waveform or a phase-modulated chaoticwaveform. Generally speaking, the term “chaotic waveform” and “chaoticsequence” may be used interchangeably.

At 1406, the method receives a data symbol to be communicated to adestination. In some embodiments, the data symbol may be a binary wordwith multiple bits. Next, at 1408 the method combines the referencechaotic waveform and the auxiliary chaotic waveform to generate acomposite chaotic data communication waveform of length equal to thereference chaotic waveform such that the temporal difference between thestarting point of the reference chaotic waveform and the starting pointof the auxiliary chaotic waveform is based on a predetermined functionor predetermined mapping of the data symbol to be communicated. Notethat the term “composite chaotic data communication waveform” is usedsynonymously with the term “composite chaotic sequence” describedearlier. In some embodiments, the reference chaotic waveform and theauxiliary chaotic waveform may be pointwise-combined additively. Inother embodiments, more than one auxiliary chaotic waveform may be usedin conjunction with the reference chaotic waveform to generate thecomposite chaotic data waveform using combinatorial methods such asaddition as described above. Finally, at 1410, the method transmits thecomposite chaotic data communication waveform to the destination.

FIG. 15 illustrates a flow diagram that describes a method 1500 forreceiving and demodulating a combination of two chaotic waveforms at areceiver associated with a chaotic communication system in accordancewith an embodiment. At 1502, the method receives a composite chaoticdata communication waveform from a transmitter. Next, at 1504, themethod demodulates the composite chaotic data communication waveformusing the reference chaotic waveform to determine the temporal positionof the corresponding correlation peak. Next, at 1506, the methoddemodulates the chaotic data communication waveform using the auxiliarychaotic waveform to determine the temporal position of the correspondingcorrelation peak. In some embodiments, the chaotic waveform demodulationfunctions may be performed using correlation functions that my includeimplementations of the matched filter. The method then proceeds to 1508where it determines the difference between the temporal position of thecorrelation peak corresponding to the demodulation of the compositechaotic data communication waveform using the reference chaotic waveformand the temporal position of the correlation peak corresponding to thedemodulation of the composite chaotic data communication waveform usingthe auxiliary chaotic waveform.

At 1510, the method recovers the data symbol encoded at the transmitterinto the composite chaotic data communication waveform based on thedifference between the temporal position of the correlation peakcorresponding to the demodulation of the composite chaotic datacommunication waveform using the reference chaotic waveform and thetemporal position of the correlation peak corresponding to thedemodulation of the composite chaotic data waveform using the auxiliarychaotic waveform. The recovery of the encoded data symbol is based on apredetermined function or predetermined mapping of the data symbolshared a priori by the transmitter and the receiver. In someembodiments, multiple auxiliary chaotic waveforms may be included in thecomposite chaotic data communication waveform. In these embodiments, thereceiver demodulates the composite chaotic data communication waveformusing each of these auxiliary chaotic waveforms, to determine theposition of each of the respective correlation peaks with reference tothe correlation peak associated with the reference chaotic waveform. Thedifference in correlation peaks is mapped to the relevant set of datasymbols as per a predetermined rule. An example of a rule that maps atemporal difference between the correlation peak associated with thereference chaotic waveform and the correlation peak associated with thedata-carrying chaotic waveform to a set of data symbols has beenpresented above in the discussion of FIG. 13.

FIG. 16 illustrates a graphical plot 1600 that illustrates two possibledegrees of freedom used to increase spectral efficiency in accordancewith an embodiment. While the description above covered the temporalshifts of an auxiliary chaotic sequence (i.e., shifts in the timedomain), in some embodiments, time-domain shifts can be performed alongwith frequency-domain shifts. In other words, the frequency of thecarrier signal associated with the auxiliary chaotic sequence may beshifted relative to the frequency of the carrier signal associated withthe reference chaotic sequence. This method further helps increase thespectral efficiency of the chaotic communication system. For example, ifc_(1k) corresponds to the reference chaotic sequence where k is the timeindex and similarly c_(2k) corresponds to the auxiliary chaoticsequence, then for a carrier frequency of f₁, the reference chaoticsequence, assuming a frequency-modulated chaotic communication system,is given by

s _(1k)=cos[2π(f ₁+α₁ c _(1k))k]

where α₁ is a scaling factor associated with the reference chaoticsequence. Similarly, the auxiliary chaotic sequence(frequency-modulated) is given by

s _(2k)=cos[2π(f ₂+α₂ c _(2k))k]

where α₂ is a scaling factor associated with the auxiliary chaoticsequence. Now, suppose f₁ is fixed and f₂ can take on F number ofdistinct values, corresponding to deliberately-induced frequency shiftsin the carrier of the auxiliary chaotic sequence. These frequency shiftscan also be referred to as frequency bins. The reference chaoticsequence and the auxiliary chaotic sequence are thus combined, with bothtime-domain and frequency-domain shifts introduced into the auxiliarychaotic sequence, and transmitted to the receiver. The receiver nowsearches over all time-domain shifts and frequency-domain shifts(frequency bins) to determine the correlation peak associated with theauxiliary chaotic sequence referenced to the time-domain andfrequency-domain correlation peak of the reference chaotic signal. For Ntime-domain shifts and F frequency domain shifts, it can be shown thatfor a base data rate of R associated with transmitting the referencechaotic sequence alone, the data rate increase due to time-domain andfrequency-domain shifts associated with the auxiliary chaotic sequenceis given by

R log₂(NF)=R(log₂ N+log₂ F)

For example, if N=1024, F=256 and R=10 kbps, then the increase in datarate using the time-domain and frequency-domain shifting techniques is

10 log₂(1024×256)=180 kbps.

This example has thus achieved an 18× increase in the data rate over thebase data rate. These additional computations would require a morepowerful processor at the receiver, but with the constant increase incomputing power, implementing these algorithms presents a small tradeoffcompared to the benefit of the increase in spectral efficiency. FIG. 16presents a plot 1600 that shows a correlation peak 1602 as athree-dimensional plot versus the two-dimensional time-domain andfrequency-domain space. The time-domain space and frequency-domain spacegive two degrees of freedom to the system designer to improve thespectral efficiency of the system. The correlation peak 1602 can now beshifted anywhere along the two-dimensional plane formed by the time axisand frequency axis. Each point in this two-dimensional plane can be madeto correspond to a data symbol. The total number of possible datasymbols that can fit into the plan depends on the granularity or theresolution along each axis. All the other techniques used to furtherincrease spectral efficiency as described above (including selecting oneor more auxiliary chaotic sequences from a larger chaotic waveformensemble) can be used along with the time-domain and frequency-domainshifting techniques described here.

This illustrates the advantage of using chaotic sequences while availingof the associated waveform diversity. For example, contemporarywaveforms such as sinusoids cannot be used efficiently for time-domainshifting since the periodicity of these waveforms introduces temporalphase ambiguities, thus reducing the total number of temporal phaseshifts possible. Binary spreading sequences cannot be generated in largeensembles as chaotic sequences can; hence the latter provides muchgreater waveform diversity.

FIG. 17 is a graphical plot 1700 that illustrates how contemporarysignal modulation techniques can be used in conjunction with chaoticmodulation techniques to further increase spectral efficiency inaccordance with an embodiment. In some embodiments, contemporarymodulation techniques such as binary phase shift keying (BPSK),quadrature phase shift keying (QPSK), quadrature amplitude modulation(QAM), M-ary phase shift keying and so on can be included in conjunctionwith the methods described above to further increase spectralefficiency. For example, returning back to the frequency-modulatedauxiliary chaotic sequence presented in the discussion of FIG. 16, BPSKmodulation can be included in the auxiliary chaotic sequence as

s _(2k)=cos[2π(f ₂+α₂ c _(2k))k+dπ]

where d=0 or 1. At the receiver, once the correlation peak isdetermined, the sign of the correlation peak can be compared, forexample, to the sign of the correlation peak associated with thereference chaotic signal. If the sign of the correlation peak associatedwith the auxiliary chaotic signal is the same as the sign of thecorrelation peak associated with the reference chaotic signal then thereceiver infers that d=1, for example. Otherwise, the receiver infersthat d=0. It is straightforward to show that using BPSK modulationincreases the value of the increase in the data rate by unity. Thus, thevalue of 18 derived above in the discussion for time-domain andfrequency-domain shifts in the description of FIG. 16 increases to avalue of 19 by including BPSK modulation. FIG. 17 shows a signalconstellation plot 1700 that includes phase-shift keying and amplitudemodulation together on the auxiliary chaotic sequence. In the examplesignal constellation plot 1700, 4 amplitude levels and 8 phase levelsare shown, for a total of 4×8=32 possible phase/amplitude points. Forexample, the four points 1702, 1704, 1706 and 1708 represent the 4amplitude levels, while the points 1708, 1710, 1712, 1714, 1716, 1718,1720 and 1722 represent the 8 phase levels. In 1700, each point in theplot is associated with an amplitude that can be one of any four values,and a phase that can be one of any eight values. The auxiliary chaoticsignal can be, for example, written as

s _(2k) =A cos[2π(f ₂+α₂ c _(2k))k+mπ/4]

where A can be any one of ¼, ½, ¾ or 1, and m can be any one of 0, 1, 2,3, 4, 5, 6, 7. Since there are a total of 32 possible phase andamplitude combinations, the corresponding increase in data rate is anadded value of log₂ 32=5 to the data rate increase factor. Thus, for theexample discussed in FIG. 16 where the increase in data rate is 18times, the inclusion of the standard amplitude and phase modulationtechniques as described here will increase the data rate increase to18+5=23 times the base data rate. Thus, a base data rate of 10 kbpsincreased to 230 kbps, a marked improvement in the spectral efficiencyof the system.

FIG. 18 presents a block diagram that depicts a system 1800 forgenerating a discrete-time frequency-modulated chaotic sequence inaccordance with an embodiment. From the basic definition presentedabove, consider a chaotic sequence c_(k), where k is a time index. Then,for a frequency f, it can be shown that the function

cos[2π(f+αc _(k))k]

where α is a scaling factor, generates a frequency-modulated chaoticsequence output. Similarly, the equation

cos[2πfk+αc _(k)]

generates a phase-modulated chaotic sequence output.

Since chaotic sequences are typically real-valued, generating thesesequences using a digital processor typically involves usingfloating-point arithmetic. The implementation of floating-pointarithmetic on a processor, whether native or emulated, is a relativelyslow process and imposes a limit on how fast chaotic sequences can begenerated in real-time. Fixed-point processing is an alternative tofloating-point processing, providing the possibility of much fastersequence generation. The challenge is to adapt the chaos-generationalgorithms to a fixed-point implementation.

Direct digital synthesizers, available both as hardware devices andsoftware cores for digital processors such as FPGAs, generate digitalsignals that are trigonometric functions (sine and cosine functions).The frequency and phase of these sine and cosine functions can becontrolled and programmed in real-time by an appropriate processingdevice such as a DSP or an FPGA, and can thus be used to generatefrequency-modulated and phase-modulated chaotic sequences. In someembodiments, a discrete-time chaos waveform generator 1802 generates areal-time chaotic sequence that is input to a frequency modulation block1806 within a direct digital synthesizer 1804. In some embodiments, thediscrete-time chaos waveform generator 1802 may be the CORDIC-basedchaos waveform generator described above. In other embodiments, thediscrete-time chaos waveform generator may be another kind offixed-point chaos waveform generator. The direct digital synthesizer1806 is configured to generate sine and/or cosine digital sequences witha fundamental frequency f, and the chaotic sequence generated by thediscrete-time chaos waveform generator 1802 frequency-modulates thesesine and/or cosine digital sequences to produce one or morefrequency-modulated chaotic signals as outputs.

The advantage of using direct digital synthesizers to generatefrequency-modulated chaotic sequences (and phase-modulated chaoticsequences) is that direct digital synthesizers are capable of generatinghigh data rate sequences in real-time, while also allowing the frequencyand/or phase of these sequences to be programmed in real-time.Furthermore, direct digital synthesizers offer wide fixed-point inputand output word widths. Using, for example, 32-bit fixed-point input andoutput words greatly increases the dynamic range of the system,providing a high quality, high rate frequency-modulated orphase-modulated chaotic sequence at the output of the direct digitalsynthesizer such as 1804. The use of a large output word width reducesthe quantization error, while the use of a large input word widthimproves the resolution of the argument of the trigonometric function,allowing for fine frequency and/or phase deviations to be programmedinto the direct digital synthesizer.

FIG. 19 presents a block diagram that depicts a system 1900 forgenerating a discrete-time phase-modulated chaotic sequence inaccordance with an embodiment. In some embodiments, a discrete-timechaos waveform generator 1902 generates a chaotic sequence that is inputto a phase modulation block 1906 of a direct digital synthesizer 1904.The chaotic sequence phase-modulates the sine and/or cosine waveformsgenerated by the direct digital synthesizer 1904 to give a discrete-timephase-modulated chaotic signal at the output of the direct digitalsynthesizer 1904, based on the discussion presented above.

FIG. 20 presents a block diagram that depicts a system 2000 forgenerating a discrete-time frequency-modulated chaotic sequence inaccordance with an embodiment. In some embodiments, a binary counter2002 of appropriate word width is used to generate a progressivelyincreasing integer value as an input to a frequency modulation block2006 of a direct digital synthesizer 2004. It can be shown that thisconfiguration can generate a discrete-time frequency-modulated chaoticsequence. For a word width of N associated with the binary counter 2002,the binary counter 2002, starting at zero, resets after it reaches acount of 2^(N)−1; however since the binary counter output is now theargument of a sine and/or cosine function, the phase of the sine and/orcosine functions at the time the binary counter resets may not be thesame as that at the beginning of the previous counter cycle. Due to thisphenomenon, it is possible that the discrete-time frequency-modulatedchaotic waveform generated during a new cycle may be entirely differentfrom the discrete-time frequency-modulated chaotic waveform generatedduring a previous cycle. The phenomenon of the phase in the sine and/orcosine functions thus may enable a chaotic sequence of a longer temporallength to be generated without repetition. The use of the direct digitalsynthesizer 2004 allows for discrete-time chaotic sequences to begenerated at high rates in fixed-point format. This technique can beused in conjunction with all the chaotic modulation and generationschemes discussed above, including frequency-modulation andphase-modulation using chaotic sequences.

The binary counter 2002 may be configured to count up in ones, or thebinary counter 2002 may be configured to count down in ones, therebygenerating different chaotic sequences. Generalizing this concept, theincrement (or decrement) in the binary counter is not limited to one; anincrement or decrement of any integer value can be used, and thisdiversity generates an entire ensemble of chaotic sequences.

FIG. 21 presents a block diagram that depicts a system 2100 forgenerating a discrete-time phase-modulated chaotic sequence inaccordance with an embodiment. In some embodiments, a binary counter2102 of appropriate word width is used to generate a progressivelyincreasing integer value as an input to a phase modulation block 2106 ofa direct digital synthesizer 2104. This configuration can generate adiscrete-time phase-modulated chaotic sequence. For a word width of Nassociated with the binary counter 2102, the binary counter 2102,starting at zero, resets after it reaches a count of 2^(N)−1; howeversince the binary counter output is now the argument of a sine and/orcosine function, the phase of the sine and/or cosine functions at thetime the binary counter resets may not be the same as that at thebeginning of the previous counter cycle. Due to this phenomenon, it ispossible that the discrete-time phase-modulated chaotic waveformgenerated during a new cycle may be entirely different from thediscrete-time phase-modulated chaotic waveform generated during aprevious cycle. The phenomenon of the phase in the sine and/or cosinefunctions thus may enable a chaotic sequence of a longer temporal lengthto be generated without repetition. The use of the direct digitalsynthesizer 2104 allows for discrete-time chaotic sequences to begenerated at high rates in fixed-point format. This technique can beused in conjunction with all the chaotic modulation and generationschemes discussed above, including frequency-modulation andphase-modulation using chaotic sequences.

The binary counter 2102 may be configured to count up in ones, or thebinary counter 2102 may be configured to count down in ones, therebygenerating different chaotic sequences. Generalizing this concept, theincrement (or decrement) in the binary counter is not limited to one; anincrement or decrement of any integer value can be used, and thisdiversity generates an entire ensemble of chaotic sequences.

FIG. 22 presents a diagram illustrating different methods 2200 ofmasking information-carrying chaotic waveforms to increase the securityof the communication system. Standard spread-spectrum systems oftentimesuse periodically repeating waveforms to transmit information. If anadversary stores long lengths of a periodically repeated waveform, itmay be possible for the adversary to auto-correlate the waveform andobserve correlation peaks that are periodic. If a noise-only waveform isauto-correlated, the correlation function will show a correlation peakonly at the initial sample. If, however, a noise-plus-spreading sequencewaveform is auto-correlated and the spreading sequence is repeated, itmay be possible to recover multiple, smaller periodic correlation peaksthat are visible on the correlation plot. These periodic correlationpeaks correspond to the spreading sequences, and give away the presenceof the spread spectrum signal, even if the spread spectrum signal isburied below the noise floor. Now, the adversary can store this waveformand use it to intercept future communications that use the samewaveform. While this process is difficult, this example shows that it ispossible to hack into spread spectrum communication waveforms.

FIG. 22 presents multiple ways in which the robustness of spreadspectrum systems using chaotic waveforms can be improved. In someembodiments, a composite chaotic waveform 2202 is comprised of twochaotic spreading sequences, a strong chaotic spreading sequencec_(s)(t) 2204, and a weak chaotic spreading sequence c_(w)(t) 2210,where the subscript “s” is used to denote a strong chaotic spreadingsequence that has a power level that is greater than the weak chaoticspreading sequence denoted by subscript “w.” The combination of thestrong chaotic spreading sequence c_(s)(t) 2204 and the weak chaoticspreading sequence c_(w)(t) 2210 is repeated periodically 2216, as shownby 2206 and 2212, and 2208 and 2214 respectively. Of the strong chaoticspreading sequence c_(s)(t) 2204, and the weak chaotic spreadingsequence c_(w)(t) 2210, only the weak chaotic spreading sequencec_(w)(t) 2210 carries any communication information. The strong chaoticspreading sequence c_(s)(t) 2204, and the weak chaotic spreadingsequence c_(w)(t) 2210 are of equal temporal length, and have theirpower levels adjusted such that the weak chaotic spreading sequencec_(w)(t) 2210 can be recovered via a matched filter correlationoperation at the receiver. In some embodiments, the weak chaoticspreading sequence c_(w)(t) 2210 is scaled down by scaling factor α. Thestrong communication signal c_(s)(t) 2204 acts as a smokescreen or asynthetic noise floor that serves to mask the weak communication signalc_(w)(t). The receiver has knowledge of both the strong chaoticspreading sequence c_(s)(t) 2204 and the weak chaotic spreading sequencec_(w)(t) 2210, and can recover the communication information containedin the weak chaotic spreading sequence c_(w)(t) 2210. If an adversaryintercepts this signal and attempts to perform an auto-correlationoperation on this signal, the adversary will observe strong correlationpeaks associated with the strong chaotic spreading sequence c_(s)(t)2204; the correlation peaks associated with the weak chaotic spreadingsequence c_(w)(t) 2210 will be masked by the strong correlation peaks ofthe strong chaotic spreading sequence c_(s)(t) 2204. Thus, even bystoring a length of the communication sequence, an adversary cannotrecover the information content in the communication signal.

In other embodiments, a composite chaotic communication waveform 2218 isconstructed similar to the composite chaotic communication waveform2202, with the periodic strong chaotic spreading sequence c_(s)(t) 2204being replaced by an aperiodic strong chaotic spreading sequencec_(s)(t) 2220. In some embodiments, aperiodic strong chaotic spreadingsequence c_(s)(t) 2220 is a chaotic sequence that has a temporal lengththat is much greater than the length of the weak chaotic spreadingsequence c_(w)(t) 2222, where the weak chaotic spreading sequencec_(w)(t) 2222 carries communication information and is periodicallyrepeated 2228 at 2224, 2226 and so on. Here, the aperiodic strongchaotic spreading sequence c_(s)(t) 2220 serves as a synthetic noisefloor to bury and hide the weak chaotic spreading sequence c_(w)(t)2222, with the power level of the aperiodic strong chaotic spreadingsequence c_(s)(t) 2220 being greater than the power level of the weakchaotic spreading sequence c_(w)(t) 2222. In some embodiments, the weakchaotic spreading sequence c_(w)(t) 2222 is scaled down by scalingfactor α and is combined with the strong chaotic sequence c_(s)(t) 2220.

In still other embodiments, a three-level chaotic signal structure isused to construct a composite chaotic communication waveform 2230. Thecomposite chaotic communication waveform 2230 comprises a weak chaoticspreading sequence c_(w)(t) 2236, appropriately scaled down by a factorof α, combined with a periodic strong chaotic spreading sequencec_(sp)(t) 2234 of temporal length equal to that of the weak chaoticspreading sequence c_(w)(t) 2236, and an aperiodic strong chaoticspreading sequence c_(sa)(t) 2232. The weak chaotic spreading sequencec_(w)(t) 2236 and the periodic strong chaotic spreading sequencec_(sp)(t) 2234 are temporally repeated 2246 at 2238 and 2242, 2240 and2244, and so on, while the aperiodic strong chaotic spreading sequencec_(sa)(t) 2232 has a temporal length that is much greater than theperiod of the weak chaotic spreading sequence c_(w)(t) 2236. Thecombination of the aperiodic strong chaotic spreading sequence c_(sa)(t)2232 and the periodic strong chaotic spreading sequence c_(sp)(t) 2234together serve to mask the weak chaotic spreading sequence c_(w)(t)2236.

Although an embodiment has been described with reference to specificexample embodiments, it will be evident that various modifications andchanges may be made to these embodiments without departing from thebroader spirit and scope of the invention. Accordingly, thespecification and drawings are to be regarded in an illustrative ratherthan a restrictive sense. The accompanying drawings that form a parthereof, show by way of illustration, and not of limitation, specificembodiments in which the subject matter may be practiced. Theembodiments illustrated are described in sufficient detail to enablethose skilled in the art to practice the teachings disclosed herein.Other embodiments may be utilized and derived therefrom, such thatstructural and logical substitutions and changes may be made withoutdeparting from the scope of this disclosure. This Detailed Description,therefore, is not to be taken in a limiting sense, and the scope ofvarious embodiments is defined only by the appended claims, along withthe full range of equivalents to which such claims are entitled.

Such embodiments of the inventive subject matter may be referred toherein, individually and/or collectively, by the term “invention” merelyfor convenience and without intending to voluntarily limit the scope ofthis application to any single invention or inventive concept if morethan one is in fact disclosed. Thus, although specific embodiments havebeen illustrated and described herein, it should be appreciated that anyarrangement calculated to achieve the same purpose may be substitutedfor the specific embodiments shown. This disclosure is intended to coverany and all adaptations or variations of various embodiments.Combinations of the above embodiments, and other embodiments notspecifically described herein, will be apparent to those of skill in theart upon reviewing the above description.

What is claimed is:
 1. A method comprising: generating, by a processingsystem, a phase angle and a trigonometric multiplier, wherein the phaseangle and the trigonometric multiplier are associated with a chaoticsequence; generating, by the processing system using the phase angle anda trigonometric function, a sample of the chaotic sequence; andupdating, by the processing system, using the trigonometric multiplier,the phase angle, to generate an updated phase angle.
 2. The method ofclaim 1, further comprising determining, by the processing system,whether the updated phase angle is within a specified numerical range.3. The method of claim 2, further comprising modifying, by theprocessing system, responsive to determining that the updated phaseangle is not within the specified numerical range, the updated phaseangle, wherein the modification results in the updated phase angle beingwithin the specified numerical range.
 4. The method of claim 2, whereinthe specified numerical range is between −180° and 180°.
 5. The methodof claim 1, wherein the processing system includes a CORDIC block thatis configured to process the phase angle and the trigonometricmultiplier and generate the sample of the chaotic sequence.
 6. Themethod of claim 1, wherein the trigonometric function is a cosinefunction.
 7. The method of claim 1, wherein the trigonometric multiplieris an integer.
 8. The method of claim 7, wherein the trigonometricmultiplier is an odd integer.
 9. The method of claim 1 wherein theprocessing system performs computations in a floating-point format. 10.The method of claim 1, wherein the processing system performscomputations in a fixed-point format.
 11. The method of claim 10,wherein the fixed-point format is a FixN_Q format.
 12. An apparatuscomprising: a direct digital synthesizer configured to generate adiscrete-time sinusoidal signal, wherein the direct digital synthesizeris configured to receive one or more inputs that are used to program aphase and a frequency of the sinusoidal signal; and a discrete-timechaos waveform generator that is configured to generate a discrete-timechaotic signal, wherein the discrete-time chaotic signal is input to thedirect digital synthesizer, and wherein the discrete-time chaotic signalis used to program the phase or the frequency of the sinusoidal signal.13. The apparatus of claim 12, wherein the discrete-time chaos waveformgenerator is a CORDIC block configured to generate the discrete-timechaotic signal.
 14. The apparatus of claim 12, wherein the discrete-timechaos waveform generator is a binary counter.
 15. A method comprising:generating a first chaotic sequence at a first power level; modulatingthe first chaotic sequence with communication information; generating asecond chaotic sequence at a second power level, wherein the secondpower level is greater than the first power level; combining the firstchaotic sequence and the second chaotic sequence to generate a compositechaotic sequence; and transmitting the composite chaotic sequence to adestination.
 16. The method of claim 15, wherein the combination is anaddition operation.
 17. The method of claim 15, wherein the compositechaotic sequence includes a third chaotic sequence, wherein the thirdchaotic sequence is combined with the first chaotic sequence and thesecond chaotic sequence, wherein the third chaotic sequence is generatedat a third power level, and wherein the third power level is greaterthan the first power level.
 18. The method of claim 15, wherein thesecond chaotic sequence serves to mask the first chaotic sequence. 19.The method of claim 15, wherein the second chaotic sequence is repeatedperiodically.
 20. The method of claim 15, wherein the second chaoticsequence is aperiodic.